Superposition TheoremEdit
The superposition theorem is a fundamental tool in linear circuit analysis. It says that when a linear network is excited by several independent sources, the total response at any point in the network (voltage or current) is equal to the algebraic sum of the responses produced by each source acting alone while all other independent sources are suppressed (turned off). This principle rests on the linearity of the components involved and is a cornerstone of methodical, modular circuit design and analysis.
In practical terms, the theorem allows engineers to break complex problems into simpler pieces. By considering the effect of each independent source separately and then adding the results, one can avoid solving a single, potentially messy system all at once. The theorem holds for circuits containing resistors, linear capacitors, and inductors, and it remains valid when dependent sources are present (they must stay active). It does not apply to nonlinear devices such as diodes or transistors operating in nonlinear regions, where the principle of linearity breaks down.
Statement
Let a linear circuit contain several independent sources and possibly dependent sources. For any node voltage or branch current in the circuit, the total response is the sum of the responses due to each independent source acting alone (with all other independent sources suppressed) while keeping dependent sources active. In the context of alternating current (AC) analysis, the principle also applies to the phasor responses of the circuit at each frequency, with the same procedure carried out for each independent sinusoidal source.
Method
Applying the superposition theorem typically follows a standard workflow:
- Identify all independent sources in the circuit. These are sources whose values can be varied externally, such as independent voltage sources and independent current sources.
- Suppress all independent sources except one. A suppressed voltage source is replaced by a short circuit, and a suppressed current source is replaced by an open circuit. Dependent sources remain active.
- Solve the circuit to find the contribution (voltage or current) at the point of interest due to the remaining active source.
- Repeat the process for each independent source, obtaining a set of partial responses.
- Sum all the partial responses to obtain the total response.
Special care is needed when dependent sources are present. Since dependent sources depend on other circuit variables, they must remain in the circuit during all partial analyses. The individual responses are then combined to yield the final result.
Variants and extensions
- AC analysis: When dealing with multiple sinusoidal sources of different frequencies, each source can be analyzed separately in the frequency domain, and the phasor responses can be added. The time-domain response is the inverse transform of the sum of these phasor contributions.
- Transient analysis: For circuits with energy storage elements (inductors and capacitors), superposition can be applied to the linear differential equations governing the circuit, provided the circuit remains linear for the time interval considered.
- The relationship to other theorems: The superposition theorem is closely related to Thevenin's and Norton's theorems, which provide equivalent representations of linear networks. By applying superposition, one can determine the contribution of sources and then combine results to obtain equivalent short- or open-circuit representations as needed.
Limitations and caveats
- Linearity requirement: The theorem holds for linear components and sources. If the circuit contains nonlinear elements (for example, diodes, transistors in nonlinear regions, or components with nonlinear characteristics), the responses to individual sources do not simply add.
- Dependent sources: Do not silence dependent sources. They can significantly affect the partial responses and the final sum.
- Initial conditions: In some transient problems, initial energy storage and nonlinear switching can complicate the straightforward application of superposition. The technique is most robust when the circuit is linear and the sources are applied in a manner that preserves linearity throughout the analysis window.
- Practical accuracy: In real-world circuits with parasitics, non-ideal behavior, or switching actions, superposition remains a powerful theoretical tool, but one must be mindful of the limits of the linear model and of the assumptions behind idealized sources.